Embed Size (px)
Citation preview
Analysis, Design and Construction of an LLC Resonant Converter
by
Shi Pu
A thesis submitted to the Graduate Faculty of
Auburn University
in partial fulfillment of the
requirements for the Degree of
Master of Science
Auburn, Alabama
May 7, 2016
Keywords: LLC resonant converters, Soft switching
Electromagnetic design, Small-signal modeling
Copyright 2016 by Shi Pu
Approved by
Robert Mark Nelms, Chair, Professor, Electrical Engineering
John York Hung, Professor, Electrical Engineering
Michael Edward Baginski, Associate Professor, Electrical Engineering
ii
Abstract
The trend in DC/DC converters development is toward higher frequency, power density and
efficiency. The traditional hard-switched converter is limited in switching frequency and power
density. The phase-shift full-bridge PWM Zero-voltage-switching (ZVS) converter has been used
widely because of its ZVS working condition. But it has a problem due to the reverse recovery of
the diodes, which reduces its efficiency. Nowadays, the LLC resonant converter is a popular
research field to consider for increasing converter efficiency.
In this thesis, the conditions to achieve the ZVS mode of the LLC resonant converter are
studied using the method of fundamental element simplification. Six different conducting stages
have been introduced individually and analyzed. Also, the operating region of the LLC resonant
converter has been studied, along with the relationship between the input and output voltage as
affected by the load and the switching frequency.
After analyzing the DC characteristics of the LLC resonant converter, its small signal model
is presented and is developed using the method of extended describing functions. Using this
model, a voltage control was designed and implemented using an MC34066 analog control chip.
A prototype LLC resonant converter was constructed and tested in the laboratory.
iii
Acknowledgments
I would like to thank my advisor, Dr. R. Mark Nelms for his patient guidance. The most
precious thing I learned from him is the attitude toward research and life, having the courage to
explore. He let me realize that failure is not despair, but hope.
I also appreciate my committee members: Dr. John Y. Hung, Dr. Michael E. Baginski for
their help and valuable suggestions.
I am very grateful to my parents, Anshan Pu and Jianhong Li, who have been encouraging and
supporting me to pursue my goal.
iv
Table of Contents
Abstract ........................................................................................................................................... ii
Acknowledgments.......................................................................................................................... iii
List of Figures ................................................................................................................................ vi
List of Tables .................................................................................................................................. ix
CHAPTER 1. Introduction.............................................................................................................. 1
1-1. Basic principles of switch-mode power supplies (SMPS)....................................................... 1
1-2. Development of SMPS ............................................................................................................ 2
1-3. Organization of this thesis ....................................................................................................... 3
CHAPTER 2. Basic principle of LLC resonant converters ............................................................ 5
2-1. Performance comparison between MOSFETs and IGBTs ...................................................... 5
2-2. Operating principle of the LLC resonant converter................................................................. 8
2-2-1 Inverter ..........................................................................................................................11
2-2-2 LLC resonant tank ........................................................................................................ 12
2-2-1 Rectifier ........................................................................................................................ 13
2-3. Operating regions for the LLC resonant converter ................................................................ 15
2-4. Analysis of the LLC resonant converter ................................................................................ 20
CHAPTER 3. Small signal modeling for LLC resonant converters ............................................. 28
3-1. Commonly used methods for small signal modeling ............................................................ 28
3-2. Extended describing function ................................................................................................ 29
v
3-2-1 Defining extended describing function ........................................................................ 30
3-2-2 Review of Grove's method ........................................................................................... 31
3-2-3 Extended describing function for multi-resonant converters ....................................... 32
3-3. Small signal modeling for the LLC resonant converter ........................................................ 35
3-4. Stablity analysis ..................................................................................................................... 41
CHAPTER 4. Circuit Design and Construction ........................................................................... 45
4-1. Parameter selection for the LLC resonant converter ............................................................. 45
4-2. Magnetics design ................................................................................................................... 47
4-3. Circuit construction ............................................................................................................... 56
4-3-1 Reference voltage source ............................................................................................. 58
4-3-2 Variable frequency oscillator ........................................................................................ 58
CHAPTER 5. Simulation and experimental results ...................................................................... 62
5-1. Analysis of simulation results ................................................................................................ 62
5-2. Analysis of experiment results ............................................................................................... 64
CHAPTER 6. Conclusions............................................................................................................ 71
BIBLIOGRAPHY ......................................................................................................................... 73
Appendix. Computer program for small signal modeling ............................................................ 76
vi
List of Figures
Figure 1.1 Circuit diagram of an LLC resonant converter.............................................................. 3
Figure 2.1 Equivalent circuits for a MOSFET and an IGBT .......................................................... 6
Figure 2.2 The LLC resonant converter .......................................................................................... 9
Figure 2.3 Equivalent circuit model of the LLC resonant converter ............................................ 10
Figure 2.4 Equivalent circuit of the inverter part .......................................................................... 11
Figure 2.5 Equivalent circuit of the resonant tank ........................................................................ 13
Figure 2.6 Equivalent circuit for the LLC resonant converter ...................................................... 14
Figure 2.7 DC characteristics for the LLC resonant converter ..................................................... 14
Figure 2.8 Operating regions for the LLC resonant converter ...................................................... 17
Figure 2.9 Relationship between Rlb and k ................................................................................... 18
Figure 2.10 Simulation of an LLC resonant convertrer in SIMPLIS ........................................... 19
Figure 2.11 Current waveform in D1 and D2 ............................................................................... 20
Figure 2.12 Waveform during operation ...................................................................................... 21
Figure 2.13 Operating circuit for an LLC resonant converter ...................................................... 22
Figure 3.1 Small signal equivalent model..................................................................................... 30
Figure 3.2 Small signal equivalent circuit .................................................................................... 36
Figure 3.3 Block diagram for the system ...................................................................................... 41
Figure 3.4 Bode plot of the open loop transfer function ............................................................... 43
Figure 3.5 Poles and zeros of the open loop transfer function...................................................... 43
vii
Figure 3.6 Open loop magnitude-phase curve .............................................................................. 44
Figure 4.1 Examples of the winding breadth b and the number of turns per section ................... 49
Figure 4.2 Transformer diagrams ................................................................................................. 51
Figure 4.3 Magnetic inductance Lm=52.3μH ............................................................................... 52
Figure 4.4 Leakage inductance Ll=3.18μH .................................................................................. 53
Figure 4.5 Resonant inductance Lr=4.57μH ................................................................................. 54
Figure 4.6 Leakage inductance of the secondary side .................................................................. 55
Figure 4.7 Circuit diagram of the prototype LLC resonant converter .......................................... 56
Figure 4.8 Control block diagram ................................................................................................. 57
Figure 4.9 Additional capacitor between gate and source ............................................................ 57
Figure 4.10 Inner structure of MC34066 ...................................................................................... 58
Figure 4.11 Oscillator and one-shot timer .................................................................................... 59
Figure 4.12 Timing waveform at 800 ns dead time ...................................................................... 60
Figure 4.13 Driving and analog control circuitry for the prototype converter ............................. 61
Figure 5.1 Simulation diagram for the LLC resonant converter in SIMPLIS .............................. 62
Figure 5.2 Simulation results from SIMPLIS ............................................................................... 63
Figure 5.3 Prototype of the LLC resonant converter .................................................................... 65
Figure 5.4 Output voltage test for the converter ........................................................................... 66
Figure 5.5 Current in the resonant tank ........................................................................................ 66
Figure 5.6 Load change from 8.15 Ω to 12.5 Ω ............................................................................ 68
Figure 5.7 Load change from 12.5 Ω to 8.15 Ω ............................................................................ 69
viii
Figure 5.8 Input voltage change from 36 V to 38.4 V .................................................................. 69
Figure 5.9 Input voltage change from 39 V to 37 V ..................................................................... 70
ix
List of Tables
Table 2-1Physical causes and designations of a MOSFET and an IGBT parasitic elements ......... 7
Table 4-1Parameters for Litz wire selection ................................................................................. 42
Table 4-2 Capacitors and resistors used for external circuitry of the MC34066 .......................... 42
Table 5-1Characteristics when input voltage remains constant .................................................... 55
Table 5-2 Characteristics when the load (output power) remains constant .................................. 55
1
CHAPTER 1. Introduction
The main goal of power electronics is to apply power electronic devices and control
technologies to manage and convert electric energy. Being a combination of electronic
technology, control technology and power technology, power electronics has great technical and
economic significance.
1-1. Basic principles of switch-mode power supplies (SMPS)
Power converters can be divided into four types:
Rectifiers (AC/DC): A rectifier converts an AC voltage to a DC voltage. Rectifying
means the power flows from the AC side to the DC side.
Inverters (DC/AC): An inverter converts a DC voltage to an AC voltage. For example, an
uninterruptible power supply can provide an AC voltage to a load from the DC stored in
batteries.
AC/AC converter: it can change both the frequency and magnitude of its AC input
voltage.
DC/DC converter: In a dc-dc converter, the dc input voltage is converted into to a dc
output voltage at another level [1].
1-2. Resonant mode conversions
Interest in switched-mode power supplies (SMPS) has been skyrocketing during the past few
decades. Resonant converters have been utilized widely in power electronics industry till now.
Over the years we have seen power conditioning move from simple but extravagant linear
2
regulators, through early low frequency pulse-width modulated systems, to high frequency
square wave converters which pack the same power handling capabilities of earlier designs into a
fraction of their size and weight. Today, a new approach is upon us the resonant mode converter.
While offering new benefits in performance, size, and cost, this new technology brings with it an
added dimension of complexity [2]. There are several commonly seen and used resonant
converters, but in this thesis we focus on the LLC resonant converter.
Series resonant converter: A series resonant converter has a resonant tank formed by a
resonant capacitor and a resonant inductor. This circuit and its derivatives are among the
most simple and least costly to produce converter circuits, but it has some disadvantages
as it cannot operate when the circuit is unloaded.
Parallel resonant converter: The characteristics of the parallel resonant converter are quite
different from those of the series resonant converter, and from those of conventional
PWM converters [2]. It has a resonant tank formed by a resonant capacitor and a resonant
inductor that are parallel connected. The voltage gain during its operating process can be
either larger or smaller than 1, which makes it popular for applications that requires
variable output voltage. However it has a relatively much higher power loss than other
resonant converters.
LLC resonant converter: The traditional topologies of resonant conversion can optimize
performance at one operating point, but not with wide range of input voltages and load
power variations. LLC resonant converter can operate over a larger range of input voltage
and switching frequency. Also, in a high voltage mode, the LLC resonant converter has a
3
higher efficiency than SRCs and PRCs.
Fig 1.1 Circuit diagram of an LLC resonant converter
1-3. Organization of this thesis
This thesis is organized as follows:
The fundamental characteristics of LLC resonant converters are illustrated in Chapter 2. An
equivalent circuit is presented, and its mathematical model is constructed. Six different operating
stages are illustrated and certain operating regions will be discussed. Also, properties of
MOSFETs and IGBTs are discussed. This is key in the selction of the semiconductor switches for
the LLC resonant converter.
The small signal modeling of the LLC resonant converter is presented in Chapter 3. Several
methods are briefly introduced, and the method of extended describing function is described in
detail. Applying the extended describing function method to LLC resonant converter yields a
small signal model. A control strategy is also discussed.
Chapter 4 is mainly focused on the design and construction of an open-loop converter. Litz
4
wire was utilized in the construction of the transformer in the LLC resonant converter. Circuit
parameters were chosen based on the characteristics discussed in chapter 2. Also, an
analog-control chip is selected to regulate the output voltage of the LLC resonant converter.
Test results for the prototype converter are presented in Chapter 5.
Conclusions and suggestions for future work are given in Chapter 6.
5
CHAPTER 2. Basic principle of LLC resonant converters
In recent years, LLC resonant converters have been widely used in power grids, renewable
energy systems and common appliances. As mentioned in Chapter 1, SRCs and PRCs have
disadvantages like power loss in resonant tanks. In 1990s, researchers have proposed
multi-resonant converters like LCC and LLC resonant converters. An LLC resonant converter
can operate stably when the input voltage and load vary over a wide range. Also, its switches
operates in ZVS mode, which can reduce the power loss during conduction. Even compared to
phase-shifted full-bridge converters, which are commonly seen in power systems, the LLC
converter has higher efficiency due to its being free from the reverse recovery problem of diodes
[3]. As a result, LLC resonant converters are commonly used in distributed power systems to
maintain the stability of the power grid.
2-1. Performance comparison between MOSFET and IGBT
MOSFETs and IGBTs are widely used in high frequency power converters. As majority
carrier devices, they are free from charge storage effect and have a high switching speed. In
addition, they have high input impedance and low driving power at the same time. MOSFETs are
more appropriate for high-frequency switch-mode power supplies, because they have a relatively
higher operating speed than IGBTs.
Figure 2.1(a) shows the internal structure of both a MOSFET and an IGBT. Figure 2.1 (b)
shows the equivalent circuits of both. Compared to a MOSFET, the IGBT has an extra layer P+,
leading directly into the collector [4].
6
Fig 2.1(a) Internal structures for a MOSFET and an IGBT[5]
Fig 2.1(b) Equivalent circuits for a MOSFET and an IGBT[5]
The physical causes and designations of the parasitic capacitances and resistances shown in
Figure 2.1 are evident in Table 2.1(a) and Table 2.1(b)
7
Table 2.1(a) Physical causes and designations of a MOSFET parasitic elements[5]
Table 2.1(b) Physical causes and designations of an IGBT parasitic elements[5]
During the switching process, the output capacitor is mainly formed of the Miller
capacitance Cgd whose intensity is determined by the reverse transfer capacitance Crss which is
equal to Cgd [4]. In a certain operating region, higher Crss will result in more intense Miller
Effect and higher output capacitance. In a MOSFET, Crss is only determined by Cgd, which
depends on the structure. Nevertheless, due to the additional P+ layer of IGBT, there is a CPN
8
formed between the P+ layer and N- layer (formed by both barrier capacitance and diffusion
capacitance). The equivalent capacitance of the IGBT is the series capacitance of Cgc and CPN,
making it smaller than a MOSFET.
For an IGBT, electrons flow through the drift region and enter the P+ region, leaving holes
(positive charge carriers) flowing into the N- region. These injected holes flow through the MOS
drain and N well into the emitter. Unlike a MOSFET, after the emitter current has stopped, many
p-charge carriers generated by injection from the IGBT collector area are still present in the
n-drift area. They must recombine or be reduced to zero by backward injection, which causes a
more or less strong collector tail current [5]. In conclusion, a MOSFET has a bigger output
capacitance, while the IGBT has a problem of tail current.
From the analysis above, it can be seen that IGBT has a relatively smaller output capacitance,
and it stores less energy during the off-stage, which leads to less turn-on power loss. As a
unipolar device, a MOSFET can take away charge on the input capacitor (Cgs+Cgd) instantly,
accelerating the turn-off process. The IGBT will suffer the tail current problem causing a turn-off
power loss.
Hence, circuits using MOSFETs as switching components should operate in the ZVS mode.
In that case, before turn on, the voltage between drain and source can be zero for a low turn-on
loss. Circuits using IGBTs should operate in the ZCS mode, having a zero current before turn-off,
lowering the turn-off loss by the tail current.
9
2-2. Operating principle of the LLC resonant converter
The main circuit of the LLC series resonant converter is shown in Fig 2.2. MOSFETs are
used in this full-bridge circuit.
Fig 2.2 Main circuit of LLC series resonant converter
T1 and T4 share the same driving signal, and T2 and T3 share the same one. A small period
of dead time between the drive signals is necessary to prevent a short circuit. Lr, Cr and Lm form
the resonant tank. The secondary side is a center-tapped rectifier followed by a capacitive filter
[7]. Diodes D1 and D2 form a full-wave rectifier circuit. Cf is part of the output filter. Under
normal circumstances, the value of Lm is relatively high so as to play a role in filtering, replacing
filter inductances on the secondary side.
If we focus on the fundamental harmonic, it will be much simpler and convenient to analyze
the converter. By calculating the ratio and relationship between the first harmonic term and
higher order terms, we can get the condition of ignoring higher order terms and simplify the
analysis process. Thus, making assumptions that all the switches and diodes are conducting
10
properly and convert the load to the primary side, we get the equivalent circuit with only
resonant components and load. Fig 2.3 shows equivalent circuits for the LLC resonant converter.
Le and Re represent the equivalent resonant inductance and resistance of the model. The LLC
resonant converter operates with a switching period of Ts, switching angular frequency of ωs,
and a resonant angular frequency of ωr. When the switches are operated properly, we can
simplify the converter circuit in Fig 2.2 to the equivalent circuit shown in Fig 2.3(a). The
equivalent circuit in Fig 2.3(a) can be redrawn as Fig 2.3(b)
(a) (b)
Fig 2.3 Equivalent circuit model of LLC resonant converter
Combining impedance ωsLm and R, yields:
222
22
222
2
;ms
mse
ms
msre
LR
LRR
LR
LRLL
(2-1)
The Fourier expansion for the input voltage can be expressed as (2-3):
1
)s i n (14
)(n
sins tnn
Vtv
(2-2)
Also, we can determine the impedance of the equivalent circuit in Fig 2.3(b):
)1
()(rs
esesinCn
LnjRjnZ
(2-3)
11
Since the quality factore
er
R
LQ
, and
re
rCL
1 , we can get the current value at
different harmonics and determine the relationship between different harmonics ( nI is the nth
harmonic of the current, and 1I is the fundamental component):
2
22
2
2
222
2
2
2
2
22
1)1(1
)1(1
n
Qnn
Q
I
I
Z
VI
r
r
s
s
r
r
s
n
in
sn
n
(2-4)
It can be seen from (2-5) that when Q is relatively big, ωs approximately equals ωr, and the
current is dominated by the first harmonic. Thus, when analyzing the large-signal model, we can
focus on the fundamental harmonic [1].
2-2-1 Inverter
The inverter is in block in Fig 2.2. We can replace the output Vs as a controlled voltage
source whose value is determined by Vin. The equivalent circuit of the inverter is shown in Fig
2.4.
Fig 2.4 The equivalent circuit of the inverter
12
Being inductive, the resonant tank has an inductor current whose phase lag is φ compared to
the input voltage. The fundamental harmonic of the resonant current is:
)s i n ()( 11 tIti sss (2-5)
Thus, the average current, the controlled current source in equivalent circuit of Fig 2.4 is:
2/
0
11 c o s
2)s i n (
2 sTs
ss
s
in
IdttI
TI
(2-6)
Since tV
tv sin
s
sin4
)(1 (2-7)
It can be included that:
cos2 1sin
outin
IVPP (2-8)
2-2-2 LLC resonant tank
The resonant tank is block in Fig 2.2 with the load converted to the primary side. Note
that the resonant tank is connected between the input voltage and the output voltage. Thus, we
can replace the resonant tank by a transfer block H(s). An equivalent diagram for the resonant
tank is shown in Fig 2.5. The crucial problem part is the derivation of the transfer function H(s),
which is given in (2-9).
(a)
13
(b)
Fig 2.5 Equivalent circuits for the resonant tank
RsLLLRCsLLCs
LRCs
sLR
sRL
sCsL
sLR
sRL
sV
sVsH
mmrrmrr
mr
m
m
r
r
m
m
s
o
)(1)(
)()(
23
2
(2-9)
2-2-1 Rectifier
The rectifier is block in Fig 2.2. Assume that the filter capacitor is large enough that it can
absorb all the AC part of the output current. Also, assume that the transformer is ideal with a
voltage ratio of N:1:1. The signal on the primary side is a sinusoidal, and the currents of D1 and
D2 flow alternately. Because the load is modeled as a pure resistance, the phase difference
between vp1 and ip is 0 where vp is the voltage across the primary side of the transformer and ip is
the current flow in the primary side of the transformer.
)s i n ()( 1 tIti spp (2-10)
When ip>0, D1 is on and Vp = NVo. When ip<0, D2 is on and Vp = -NVo. Thus, we can get
the Fourier series for the primary side voltage:
R
14
....5,3,1
)(sin14
)(n
nso
p tnn
NVtv
(2-11)
Its fundamental component is )sin(4
)(1
tNV
tv so
p (2-12)
Since 11
1 4
)sin(
)sin(4
)(
)(
p
o
sp
so
p
p
I
NV
tI
tNV
ti
tvR
(2-13)
Also, the average output current is
12/
01
2)sin(
2 pT
sp
s
o
NIdttNI
TI
s
(2-14)
We know that 12 p
o
o
oL
NI
V
I
VR
(2-15)
Therefore, LRN
R2
28
(2-16)
As a result, L
p
outin RIN
PP2
2
1
24
, which assumes an efficiency of 100%.
Now that we have analyzed each part of the circuit in Fig 2.2, we can cascade them together
to form an equivalent circuit for an LLC resonant converter based on the fundamental
approximation.
Fig 2.6 Equivalent circuit for an LLC resonant converter
From the equivalent circuit we know that the step-up ratio M is:
15
N
jHjH
RN
NR
V
VM
s
s
L
L
in
o)(4
)(8
22
2
(2-17)
Using MATLAB, we can plot the DC voltage characteristic of the LLC resonant converter as
shown in Fig 2.7.
Fig 2.7 DC characteristics of the LLC resonant converters
From Fig 2.7, it can be seen that M changes with frequency. Also, there are two different
resonant frequencies denoted as fr and f1. When the load becomes larger (Q becomes larger), the
frequency where the peak value occurs increases.
2-3. Operating regions for the LLC resonant converter
When the MOSFETs in Fig 2.2 are driven by complementary symmetric signals and the
shunt capacitance and the resonant capacitance are quite different, the conditions for ZVS and
DC characteristics
Voltage
Gain
(M)
fs/fr
Q=0.3
Q=6
Q=3.6
Q=1.6
Q=1
Q=0.58
Q=0.4
Q=0.35
Q=0.50
fr f1
16
ZCS are mainly determined by the input impedance of the resonant tank. When inZ is inductive,
the input current ir lags behind the input voltage vs by ϕ degrees. Thus, before the switches are
turned on, the parasitic diodes are already conducting. As a result, the voltage across the switches
is fixed to zero, realizing ZVS operation. However, when the driving signals are removed,
current is still flowing in the switches, which means the turn-off stage is a hard shutdown. In
contrast, when inZ is capacitive, the input current ir leads the input voltage vs by ϕ degree. Thus,
before the switches are turned on, the shunt diode of the other switch from the same bridge is
conducting. As a result, the voltage across the switches is fixed to Vin, which means it’s a hard
turn on. However, when the driving signal is removed, the resonant current flows through the
parasitic diode resulting in ZCS mode.
Once the load and circuit parameters have been determined, the impedance inZ is
determined and can’t be changed. Thus, an LLC resonant converter doesn’t have the ability to
switch its operating mode between ZVS and ZCS. Since the imaginary part of the input
impedance inZ determines whether it’s capacitive or inductive, let’s examine the imaginary part
first.
24224
24
64
641))((
Lms
smL
rs
rssinmRNL
LRN
CLjZI
(2-18)
When ωs<ωr, the zero-crossing frequency of Im( inZ ) increases as the RL decreases, leading
to its boundary ωs=ωr. When ωs>ωr, the imaginary part is always positive. We can divide the DC
characteristic diagram into three parts as shown in Fig 2.8 [6]. Since we already know that the
circuit will operate in ZVS mode when inZ is inductive, we define this certain area to be Region
17
2. Conversely, we define the area where inZ is capacitive and the circuit operates in ZCS mode
to be Region 1. As seen in Fig 2.8, Region 2 includes two different parts. One is the area where
ω1 < ωs < ωr and another one is ωs > ωr. The first area is labeled as Region 2.1, and the second one
is Region 2.2.
Fig 2.8 operating regions of LLC resonant converters
Because LLC resonant converters operate in a high-frequency mode, MOSFETs are widely
used. As we have discussed, when using MOSFETs, the circuit should operate in a ZVS mode to
minimize the power loss during switching. Thus, the LLC resonant converter should operate in
Region 2, instead of Region 1 [7].
When ωs < ω1, the converter will always operate in the ZCS mode. While ωs > ωr, the
converter will always operate in the ZVS mode. When ω1 < ωs < ωr, the operation depends on the
18
value of the load. Thus, we need to know the value of the load for the boundary between the ZVS
and ZCS modes. Letting Im( inZ )=0, we obtain:
064
64124224
24
Lms
smL
rs
rsRNL
LRN
CL
(2-19)
Define 1)1(
1
8 2
2
2
2
kh
k
N
khZR r
Lb
(2-20)
When RL > RLb, the converter operates in the ZVS mode, where rsk / . It can be seen
in Fig 2.9 that RLb = 0 when ωs > ωr, which means that the converter will always operate in the
ZVS mode. When ω1 < ωs < ωr, a higher switching frequency makes it easier for the converter to
realize ZVS mode.
Fig 2.9 The relationship between RLb and k
When the converter is operating, the ideal status for the rectifier diodes is that there are not
any reverse voltage oscillations. Because the converter operates in a high-frequency mode, the
RLb
ωs/ωr
19
distributed inductance of the secondary side (transformer’s leakage inductance, line inductance)
cannot be ignored. In Region 2.2, when D2 is on, D1 is reverse biased. The electric charge on the
junction capacitance cannot be released instantly, which means D1 keeps conducting. However,
at this time, D2 has already been turned-on, leaving a reverse recovery current flow in D1,
causing a voltage oscillation across D1. While, in Region 2.1, before the diode D2 is turned on,
the current in D1 has already reached zero, which means the diode has restored its reverse
blocking capability. To better illustrate the differences between these two circumstances,
simulation software SIMPLIS was used to generate simulated waveform results for these two
regions. The SIMPLIS circuit is shown in Fig 2.10.
Fig 2.10 Simulation of an LLC resonant converter in SIMPLIS
In Fig 2.11, the case where the current flows through the two diodes D1 and D2 is presented.
In Fig 2.11(a), the current flows through D1 and D2 share no overlapped area realizing a lower
power loss operating mode. On contrary, the current waveform in Fig 2.11(b) tells us when D1 is
turned on, D2 is still conducting. Thus, Region 2.1 has lower operating loss than Region 2.2.
20
Fig 2.11 (a). Current waveform in D1 and D2 in Region 2.1
Fig 2.11 (b). Current waveform in D1 and D2 in Region 2.2
Due to the power loss caused by the reverse recovery, the ideal operating region is Region
2.1, not Region 2.2.
2-4. Operating process analysis for the LLC resonant converter
In this section, we make the following assumptions: Vin is constant, Cf is large enough to fix
Vo at a constant value, and Lm is large enough to let the resonant tank complete a full resonant
period.
As discussed earlier, the LLC resonant converter is a multi-resonant converter since the
resonant frequency in different time intervals are different. Thus, the waveform should be
divided into clearly two time intervals. The corresponding waveform of each component is
shown in Fig 2.12.
21
Fig 2.12 waveform during operating [3]
The whole operating process can be divided into six different stages [8] [9], which are
shown in Fig 2.13. In these stages, t0 is the time when the current flowing through the resonant
inductor Lr reaches 0 A. The current flow through Lm equals that through Lr at t1. The resonant
current starts to decrease at t2. At t3, the resonant current reaches 0 A. The current flow through
Lm equals that through Lr (negative value) at t4 and starts to increase at t5. At t6, the full resonant
period is complete.
VT1, VT4
VT2,VT3
22
Fig 2.13(a). t0 to t1
Stage 1(t0 to t1): At t0, T1 and T4 are turned-on, the resonant current flows through T1 and
T4. Because before t0, Da and Dd were on, it’s a ZVS process. Since the voltage between the two
poles of the secondary side of the transformer is positive, D1 is turned-on, supplying power to
the load. During this period of time, the voltage across Lm is fixed at a constant value of NVo.
Thus, iLm increases linearly and Lm doesn’t participate in the resonant process. The resonant
frequency is fr (rr
rCL
f2
1 ). Applying KCL and KVL, we can calculate the resonant current
ir and the voltage across the resonant capacitor Cr. In the following stages, KVL and KCL are
utilized to calculate ir and vcr to verify the waveform in Fig 2.12. Also, the equations are
preparation for the small signal modeling in the next chapter. For stage 1,
)(c o s)()(
)(s i n)(
0
0
ttVNVVNVVtv
ttZ
VNVVti
rcrmoinoincr
r
r
crmoinr
(2-21)
23
Fig 2.13 (b). t1 to t2
Stage 2(t1 to t2): when time reaches t1, ir=iLm, due to KCL we know that the current flow
into the primary side of the transformer is 0 A. So, there is no power transferred to the secondary
side of the transformer. The voltage across Lm is no longer fixed at a certain value, and Lm will
participate in the resonance in the second time interval. Because Lm is normally very large, we
can assume that the resonant capacitor is charged constantly and its voltage increases linearly.
(Im is the maximum value of the resonant current and the resonant frequency is f1) The equations
for this stage are
)()()(
)(
2
11
12
ttC
Itvtv
Iti
TTtt
r
mcrcr
mr
rs
(2-22)
24
Fig 2.13 (c). t2 to t3
Stage 3(t2 to t3): At t2, T1 and T4 are turned-off, Db and Dc turn on. As a result, the voltage
across the primary side of the transformer is negative, turning on the diode D2. The voltage
across Lm is fixed at -NVo. Now, Lm is no longer part of the resonant tank. Its current decreases
linearly. The resonant frequency of this period is fr. The equations for this stage are
)(s i n)(c o s))(()(
)(sin)(
)(cos)(
222
22
2
ttIZtttVNVVNVVtv
ttZ
tVNVVttIti
rmrrcroinoincr
r
r
croinrmr
(2-23)
Fig 2.13 (d). t3 to t4
25
Stage 4(t3 to t4): At t3, T2 and T4 are turned on. Because Db and Dc were on, this is a ZVS
turn on. The voltage across the secondary side of the transformer is negative, and D2 continues
to conduct. The voltage across Lm decreases linearly. The equations for this stage are
)(cos))(()(
)(sin)(
)(
33
33
tttVNVVNVVtv
ttZ
tVNVVti
rcroinoincr
r
r
croinr
(2-24)
Fig 2.13 (e). t4 to t5
Stage 5(t4 to t5): At t4, the current through Lm reaches -Im. There is no power flowing into
the secondary side of the transformer. The equations for this stage are
)()()(
)(
2
44
45
ttC
Itvtv
Iti
TTtt
r
mcrcr
mr
rs
(2-25)
26
Fig 2.13 (f). t5 to t6
Stage 6(t5 to t6): T2 and T3 are turned-off, and Da and Dd turn on. The current through Lm
increases linearly. The equations for this stage are
)(sin)(cos))(()(
)(sin)(
)(cos)(
555
55
5
ttIZtttVNVVNVVtv
ttZ
tVNVVttIti
rmrrcroinoincr
r
r
croinrmr
(2-26)
In an ideal situation, there are no DC components in the resonant current and resonant
voltage. Thus crmcrcr vtvtv )3()0( . Substituting this equation to (2-21) and (2-23), we can get:
2)1()12()1()2( rs
r
mcr
r
mcrcr
TT
C
Itvtt
C
Itvtv
(2-27)
Because )01(cos)()1( ttVNVVNVVtv rcrmoinoincr (2-28)
After substituting (2-28) into (2-27), and applying KVL due to the assumption that
mItiti )2()1( , we can get
)2()(s i n)1( tittZ
VNVVti or
r
crmoin
(2-29)
Substitute (2-29) to the conclusion of (2-21), we obtain:
)0(s i n)(*)1( ttVNVVZti rcrmoinr (2-30)
27
Because the characteristic impedance rr
rC
Z
1 , while the impedance of Cr at resonant
frequency isrrC
j
1 . (2-30) can be converted into the voltage across the Cr at time t2 as
)0(cos)( ttVNVV rcrmoin . Thus, substituting v(t2) and (2-28) into (2-27) we can get
r
mrsino
NC
ITTVNV
4
)(0
(2-31)
rsr
msrino
ffNC
Iff
N
VV
4
)( (2-32)
From the operating process we can see that there is no power switching during stages b and e,
which is the main reason influencing the efficiency of the circuit. Also, from (2-28) we can see
when Vo is fixed, Vin and Ts are negatively correlated. In other words, when the output voltage is
higher, the efficiency becomes larger.
In this chapter, we compared two commonly used semi-conductor switches MOSFETs and
IGBTs at first. After analyzing their features, we picked MOSFETs as the switches of this LLC
resonant converter. Subsequently, we studied the LLC resonant converter in separated stages and
obtained its large signal model and the voltage ratio of the input and output voltages. As a result,
by utilizing MATLAB, we determined the operating region of the converter. And finally, based
on the chosen operating region, we analyzed the different stages and obtained the expression of
the output voltage and learned more about LLC resonant converter’s features. In order to
construct a stable LLC resonant converter, we also have to obtain its small signal model to
analyze its stability and characteristic. This part will be discussed in the next chapter.
28
CHAPTER 3. Small signal modeling for LLC resonant converters
The small signal model of a SMPS system can represent the behavior of the converter near
its equilibrium or stable operating point. In this chapter, several commonly used methods are
introduced. Later the method of extended describing function is presented and utilized to model
the LLC resonant converter. At last, the stability of the system is checked by utilizing computer
program developed in MATLAB.
3-1. Commonly used methods for small signal modeling
For switching power supply control, it’s often not easy to determine a small signal model in
the presence of non-linear properties [10], [11], [12]. It shows the non-linearly property for two
reasons:
1) switching process: When MOSFETs are turned on and off at different times, the circuit
topology changes. The state variable equations describing the circuit operation vary periodically.
2) components: Component non-linearity results from the saturation of magnetic cores.
There are several ways to develop a small signal model.
State-space averaging method: Ignore the higher order terms in one period when the
switching frequency is high. Use a linear time invariant (LTI) state equation to
approximate the operation of the SMPS system.
Generalized averaging method: Based on a harmonic balance, Fourier analysis can be
used on each state variable to derive a small signal model.
Discrete time-domain simulation method: Use the state-space method to infer piecewise
linear equations and non-linear differential equations.
29
Dr. Eric X.Yang has developed a small signal modeling process based on the generalized
averaging method [10]. For this method, apply Fourier analysis to each state variable and obtain
a small signal model using the harmonic balance method, which was proposed by J. Groves [11].
This modeling method is referred to as Extended Describing Function and can be utilized in the
analysis and modeling of resonant and multi-resonant converters.
3-2. Extended describing function
For DC/DC converters, an equivalent system contains three individual input variables
known as the controlling input (switching frequency), input voltage (Vg) and loading current (Io).
Variables )(ˆ),(ˆ),(ˆ),(ˆ sisVssV ogso stands for the small signal perturbation values. The system can
be described by the following equivalent model:
)(ˆ)()(ˆ)()(ˆ)()(ˆ sisGsVsGssGsV ovigvgsvwo s (3-1)
In this model,
0)(ˆ0)(ˆ
0)(ˆ0)(ˆ
0)(ˆ0)(ˆ )(ˆ
)(ˆ)(;
)(ˆ
)(ˆ)(;
)(ˆ
)(ˆ)(
s
sVo
ovi
si
sg
ovg
si
sVs
ovw
s
g
o
s
o
g
s si
sVsG
sV
sVsG
s
sVsG
,
stands for transfer function from switching frequency, input voltage and load current to the
output voltage. A system model based on (3-1) is shown in Fig 3.1
30
Fig 3.1 Small signal equivalent model
3-2-1 Defining extended describing function
The state equation of the converter system can be written as:
),,(
),,(
tuxgy
tuxfx (3-2)
Under steady-state, the input u is a constant vector and the state trajectory is periodic over the
switching period, Ts. Therefore, the state vector can be expanded into Fourier series:
k
tjkss
k
ss seXtx
)( (3-3)
where
s
s
Ttjkss
s
ss
kT
dtetxT
Xs
s
2,)(
1
0
(3-4)
The superscript, ss, represents the steady-state operation.
Similarly, the nonlinear function defined in (3-2) can also be expanded into Fourier series
under steady state operation:
)(ˆ sVin
)(ˆ s
)(ˆ sVo
)(ˆ sIo
31
k
tjk
o
ssss
ko
ss dteUXFtUxf s),(,, (3-5)
where
,...),...,,...,(...,
),),((1
),(
0
0
ss
k
ssss
k
ss
Ttjk
o
ss
s
o
ssss
k
XXXX
dtetUtxfT
UXFs
s
(3-6)
The Fourier coefficient, ),( o
ssss
k UXF is a function of the steady-state input, Uo, and all of the
Fourier coefficients of )(txss. Therefore, ),( o
ssss
k UXF can be called a multi-variable describing
function [10].
3-2-2 Review of Grove’s Method [11]
As mentioned earlier, the extended describing function method is based on the harmonic
balanced method proposed by J. Grove. Thus, a brief introduction of J. Groves method is
necessary. His method is based on the Fourier analysis over the commensurate period [11]. The
bridge that relates the state vector spectrum and the non-linear function spectrum (describing
function terms) is harmonic balance. Thus, an equation can be derived as:
l
tjl
l
l
tjl
lccc eUXFeXjl
),( (3-7)
In (3-7), the term tjl ce
are the same. Thus, to let the equation to be established, the Fourier
coefficients of each term harmonics should be equal, which can be written as:
),( UXFXjl llc (3-8)
This procedure is called harmonic balance. Expanding by Taylor series, the equation can be
written as:
32
m o
ll
l
llc u
U
FX
X
FXjl ˆ (3-9)
If the partial derivatives are known and we limit the terms as a finite number from -H to H,
then the small-signal spectrum Xl can be obtained. In this way, the system’s small signal model
can be derived. Nevertheless, J.Grove’s method has some disadvantages. The harmonic balance
method depends on knowing the information of every modulation frequency in order to obtain
partial derivatives for calculating. Thus, this procedure is very complicated and inconvenient to
use.
3-2-3 Extended describing function for multi-resonant converters
The nonlinear state equation for an LLC resonant converter is given in (3-2). Applying
Fourier analysis to each side of the two equations yields:
k
tjk
k
k
tjk
k
k
tjk
k
k
tjk
k
s
s
s
s
eUXGtuxg
eUXFtuxf
etYty
etXtx
),(),,(
),(),,(
)()(
)()(
(3-10)
As mentioned, defining ),( UXFk and ),( UXGk to be describing functions of the state
variables and output variables.
oo
T
kk YYUUXXX ,,,,,0,,, stand for state variables .
x , u is the
input voltage, and y is the output DC current. Here we should note that U and Y only have DC
33
components. For each switching period, voltage across Lr and Cr is reset to 0 V along with the
current, which makes 0dt
dx.
Thus, the 0th order of the describing function ),( UXFk is 0.
Ts
s
dttuxfT
UXF0
0 0),,(1
),( (3-11)
Now take derivatives of each side of equation (3-10):
k k
tjk
ks
tjk
kss etXjKetXtx
)()()( (3-12)
Here, J. Grove’s harmonic balance method is utilized. Substituting (3-12) back into (3-2), we
obtain:
),(
),()()(
UXGY
UXFtXjktX
oo
kksk (3-13)
Equation (3-13) is actually the frequency expression for (3-2), when the system is stable,
0)(,)( tXXtX k
ss
kk ,
Equation (3-13) can be written as
),(
),()(
000 UXGY
UXFtXjkssss
o
ssss
kks (3-14)
When there are small signal perturbations,
oo
o
s
ss
s
k
ss
kk
yYy
uUu
XXtX
ˆ
ˆ
ˆ
ˆ)(
(3-15)
If we make a substitution into (3-14), expand describing functions ),( UXFk and ),( UXGk
34
into Taylor series, ignore partial derivatives higher than second order and cancel the stable
components, we can get:
H
Hm
ss
mss
m
ss
H
Hm
ss
kmss
m
ss
kks
ss
ks
ss
k
uU
Gx
X
Gy
uU
Fx
X
FxjkXjkX
ˆˆˆ
ˆˆˆˆˆ
0
000
0
(3-16)
From these equations, if the variable is only s, then the system small signal model can be
written as:
H
Hm
mss
m
ss
H
Hm
mss
m
ss
kks
ss
ks
ss
k
xX
Gy
xX
FxjkXjkX
ˆˆ
ˆˆˆˆ
00
(3-17)
This is the familiar form
xCy
BxAx
so
sss
ˆˆ
ˆˆˆ .
Here,
),...,,...,(
),...,,...,(
......
............
......
............
......
ss
H
ss
o
ss
m
ss
o
ss
H
ss
os
Tss
H
ss
m
ss
Hs
sss
H
ss
H
ss
m
ss
H
ss
H
ss
H
ss
H
ss
msss
m
ss
m
ss
H
ss
m
ss
H
ss
H
ss
m
ss
Hsss
H
ss
H
s
X
G
X
G
X
GC
jHXjmXjHXB
jHX
F
X
F
X
F
X
Fjm
X
F
X
F
X
F
X
FjH
X
F
A
(3-18)
If we examine the different states of the LLC converter, we can divide the whole switching
35
period into different pieces. Then,
uDxCy
uBxAx
iio
ii
ˆˆ
ˆ ii TtT 1 (3-19)
Q
t
tjkT
Toi
ss
i
s
o
ssss
k dteUBtxAT
UXF si
i1 1
)(1
),(
(3-20)
i
i
sis
s
T
T
tjk
ss
m
ss
iss
m
iTjk
oi
Q
t
i
ss
iss
m
iTjk
oii
ss
i
s
ss
m
o
ssss
k
dteX
txA
X
TeUB
TxAX
TeUBTxA
TX
UXF
1
1)(
]
)([])([1),(
1
1
1
(3-21)
Where Q stands for the numbers of pieces in one switching period.
As a result, tjm
ss
m
ss
seX
x
(3-22)
Assuming that the boundary condition is
0)(),,( biobiibii
ss
bio
ss
i DUCTBTxAUxTh ,
bioii
ss
ibi
Tjm
bi
i
ss
m
ss
m
i
BUBTxAA
eA
T
h
X
h
X
T is
])([
(3-23)
Due to equations (3-21), (3-22) and (3-23), we can get the small signal model of the system.
We have introduced the basic method of Extended Describing Function in this section. Since
Now we can derive an LLC resonant converter’s small signal model if we know the linearized
state equations for different operating stages. The next step is to calculate state equations of
pieces of the linear network in one switching cycle.
36
3-3. Small signal modeling for the LLC resonant converter
In this section, we regard the perturbation of the load (state variable oI ) as the only
perturbation in the converter. The equivalent series resistance (ESR) of the capacitor Cf would
cause a voltage drop on the output variable Vo; thus, a low value of ESR is considered during the
modeling process. We obtain 4 different modes of operation for the LLC converter which are
shown in Fig 3.2 (a, b, c, d). Each mode corresponds to certain operating stages in Fig 2.13.
A piece-wise linearizated equation can be derived from the small signal modeling method
described above.
In this section, we assume the input matrix T
oin IVu ],[ , output matrix T
go IVy ],[ , Ig is
the input current. State variablesT
CCLLL
T
frmmrvviiixxxxx ],,,[],,,[ 4321 .
Fig 3.2 (a) mode a (stages 1&6)
Fig 3.2 (a) shows the equivalent circuit for topology “mode a” that stands for operating
pieces of stage 1 and stage 6 in Chapter 2. In stage 1 and stage 6, power flows from the primary
side to the secondary side. In Fig 3.2, all the components on the secondary side are referred to
the primary side. Cfp represents the filter capacitor Cf on the primary side, Rcep represents the
ESR value on the primary side and R represents the load resistance value RL on the primary side.
37
Also, to introduce the variation of the load current io to the primaty side, a current source io/N is
added to the circuit’s primary side. Applying KCL and KVL to the circuit operating piece of
mode a, we can obtain the equations for the state variables.
21
41
442
414
213
32121
)(
)(
)(1
)()(
xxIN
ixCxRNV
xCRxxLN
i
R
xxx
R
CRC
xxC
x
VxxxRxLLxL
g
ofpo
fpcepm
ofpce
fp
r
inpmrr
(3-24)
Definecepcep
cep
RR
Rkand
RR
RRr
,
.
As a result:
fpfp
rr
mm
mrrr
p
mr
p
RC
k
C
k
CC
L
k
L
r
LLk
LL
R
L
r
L
Rr
A
00
0011
00
)11
(1
1
0011
001 N
k
N
r
C
T
fpmmr
r
NC
k
NL
r
LLN
r
LB
0)11
(
0001
1
00
02
1 N
r
D
38
Fig 3.2 (b) mode b (stage2)
In mode b, there is no power flow from the primary side to the secondary side. Thus, the
components in the secondary side are separated from those in the primary side, which means iLr =
iLm Applying KVL and KCL to the circuit, the equations for mode b can be derived as:
2
24
44
23
322
1
1
1
0
xI
iN
rx
N
kV
NC
ki
RC
kxx
xC
x
LL
Vx
LLx
LL
Rx
x
g
oo
fp
o
fp
r
mr
in
mrmr
p
(3-25)
fp
r
mrmr
p
RC
k
C
LLLL
R
A
000
001
0
01
0
0000
2
0011
0002 N
k
C
39
T
fp
mr
NC
k
LLB
000
001
0
2
00
02
2 N
r
D
Fig 3.2 (c) mode c (stages 3&4)
Compared to stage 1 and stage 6, mode c has the same topology mode that shows the power
flows from the primary side to the secondary side. But the difference between mode a and mode
c is that Vin, Vo and the voltage across the filter capacitor has the opposite polarity. Thus, the
matrices can be easily derived.
For mode c:
fpfp
rr
mm
mrrr
p
mr
p
RC
k
C
k
CC
L
k
L
r
LLk
LL
R
L
r
L
Rr
A
00
0011
00
)11
(1
3
0011
003 N
k
N
r
C
40
T
fpmmr
r
NC
k
NL
r
LLN
r
LB
0)11
(
0001
3
00
02
3 N
r
D (3-26)
Fig 3.2 (d) mode d (stage 5)
Matrices for mode d can be derived by changing the direction of Vin, Vo and Vcfp of
matrices for mode b. In other words, mode c and mode a share the same eigenvalues, while mode
b and mode d share the same eigenvalues. For mode d:
fp
r
mrmr
p
RC
k
C
LLLL
R
A
000
001
0
01
0
0000
4
0011
0004 N
k
C
T
fp
mr
NC
k
LLB
000
001
0
4
00
02
2 N
r
D (3-27)
Now we have information of four operating pieces in one switching period. As introduced
41
earlier that with state equations in each of these pieces, equation (3-21) can be derived by
substituting these state equations into Ai, Bi, Ci and Di (i=1,2,3,4). Note that this procedure is
complicated so a computer program based on MATLAB should be utilized. The software
package for extended describing function can be found in dissertation of Dr. Eric X. Yang[10].
3-4. Stability analysis
The block diagram of the LLC converter controlled system is shown in Fig 3.3:
Fig 3.3 Block diagram for the control strategy
In Fig 3.3, Gvio is the transfer function of the resonant tank. Also, when adding the feedback
to the system, the comparison result of the output voltage and the reference voltage has to be
converted into a varying modulating frequency. Thus, Gvco represents the transfer function of
the voltage controlled oscillator (VCO), and H(s) is the sampling circuit’s transfer function. In an
ideal situation, VCO is assumed as a linear function whose input is the voltage control signal Vc
and the output is the switching frequency f.
In order to illustrate the stability of the feedback controlled LLC resonant converter, Matlab
is used to calculate the small signal model of the converter and utilize control theory to test the
stability of it. In the simulated system, parameters utilized are shown as follows:
s
42
nFCuHLuHL
rruFCkHzf
LLhNVV
rmr
cepfr
rmin
470; 5.52; 8.7
; 01.0; 20; 80
;6/;4; 40
After substituting parameters into the small signal model which was proposed in the
former section, state space matrices A,B,C and D were calculated in Matlab and the transfer
function was derived as well. The MATLAB file is given in Appendix A.
The transfer functions for each block shows as follows:
310*20)(;11
5)( sGsH vco
7351624917.23255-0074426549950.8276
004468092127659.574468092127659.57
33114919028.886700331148190.288867
583573128739.246-381443128865.979-3814431288.65979-9650172576.05225-
A
74426549950.8276
0
331149190.288867
5835731287.39246-
0
0
0
381443128865.979
B
0011
0.999000.00999C
00
01.00D
2015211334
1629344
10*69.210*4.110*81.210*49.7
25.389369.1*8.210*996.401.0)(
ssss
sesesssGvio
In order to test the stability of the system. Bode plot of the transfer function was plotted
in Fig 3.4. The system is a high order system that has several zeros and poles, which are plotted
in Fig. 3.5. In this figure, the zeros are relatively closer to the original. For verifying the stability,
43
a Nyquist diagram is necessary. The magnitude phase curve of the system open loop transfer
function is shown in Fig. 3.6.
Fig 3.4 Bode plot of the open loop transfer function
Fig 3.5 Poles and zeros of the open loop transfer function
44
Fig 3.6 Open loop magnitude-phase curve
We use the Nyquist stability criterion to check stability. The number of poles of
open-loop transfer function in the RHP is P=0. Also, the number of times that the
magnitude-phase curve circles (-1,j0) in the counterclockwise direction is N=0. Thus, number of
the closed-loop function’s positive real eigenvalue is Z=0. As a result, the system is stable.
In this chapter, the introduction of the extended describing function is illustrated firstly.
Because the method of extended describing function requires the piece-wise linear functions of
each operation mode, the linearized matrices for Region 2.1 were presented. Finally, by utilizing
the method of extended describing function, the stability of the system was verified. Now the
design of a prototype converter is described in Chapter 4.
45
CHAPTER 4. Circuit Design and Construction
In this chapter we select the parameters for the prototype converter. And then the process of
constructing an LLC resonant converter is illustrated. At last, the appropriate analog-control chip
is utilized to control the output voltage.
4-1. Design of the LLC resonant converter
The design parameters for the converter are:
Input voltage: 35~40 V;
Output voltage: 10 V(+/-5%);
Rated output current: 1 A;
Rated power: 40 W;
Switching frequency range: 60 kHz~80 kHz.
In this circuit, the first parameter to be selected is the turns ratio of the transformer. Since the
input voltage varies between 35 V to 40 V, and the output voltage is about 10V, the ratio of the
input voltage to output voltage is 3.5 to 4. Since the voltage gain M of the converter is bigger
than 1 when operating in Region 2.1. we use a transformer ratio equals to 4 to satisfy the
requirement. Also, Lr and Lm are realized with discrete inductors.
The resonant capacitor has been used not only to eliminate the DC current, but also as a part
of the resonant tank. Because the resonant energy is determined by the output power. We can
obtain the information of Cr from the output voltage. Here an assumption is made to approximate
the charging process of Cr. For Icr = Io/N, we can assume that in half of the switching period, Cr
has been linearly charged from -Vcrm to +Vcrm (Vcrm is the maximum voltage across Cr). Since the
46
charging time is half of the switching period, we can get:
r
soc r mcr
NC
TIVV
22 max
In this prototype converter, as long as the capacitor’s value can meet the demand of safely
operating, the capacitor can be selected. Thus, let’s choose Cr = 470 nF, Vcrm = 15.17 V. The
ceramic capacitor we utilize has a voltage limit of +/- 25 V. Thus, the 470nF capacitor can
operate safely.
Then: HCrf
Lrr
43.84
122
The value of h is determined (h=Lm/Lr) by the switching frequency range. As discussed earlier,
rs
msrino
fNCrf
Iff
N
VV
4
(4-1)
Based on circuit analysis, the voltage across Lm is NVo, and its charging time is half of the
resonant period Tr/2 as discussed in Chapter 2. Thus, the current flow through Lm increases
linearly. The maximum value of the resonant current can be derived as
m
rom
L
TNVI
4 (4-2)
Substituting (4-2) into (4-1), we can get:
)1(4
2
s
roino
f
f
h
V
N
VV
(4-3)
As Vin = 40 V, we can substitute Vin into (4-3) and with the assumption that the Lm is large
enough to maintain the resonant current value (ILm = ILr) in the second time interval at a fixed
47
value. we can get 2
2
min
2
hf
f
r
s. In Dr. Bo Yang’s dissertation [3], a conclusion was
discovered that when h is relatively large, the converter has better performance but it degrades
very fast as input voltage drops. When h is relatively small, the converter has more balanced
performance for the whole range of input voltages, but the performance at high input voltage is
greatly impaired. A common experience value range for h is between 3 to 7. Here, let’s select h
to be 6.76 and the Lm is set to be 55.4 μH.
As mentioned in Chapter 2, the loading resistance boundary between ZVS and ZCS is:
1)1(
1
82
2
2
2
n
nnrlb
hN
hZR
(4-4)
Therefore, we substitute the transformer ratio N, resonant capacitor value Cr and the value of h
into (4-4) and obtain Rlbmax=1.43 Ω, For Rlmin, 40/1=40 Ω. So, this converter will operate in ZVS
switching mode over the full load range. The maximum ripple voltage on the output was set to be
10% of the rated output voltage and is 1V. It is complicated to determine the uncharged time for
the filter capacitor because it’s not only uncharged during stages 3 & 6 when there is no power
flows to the secondary side, but also in other stages when the current flow through the filter is
larger than Io. Let’s assume that the discharging time for Cf is Ts to make sure that the capacitor
will surely satisfy the ripple requirement. Since the capacitor discharges at a constant value of Io
and the discharging time is set to be Ts, the value of Cf can be calculated as follows:
F
U
TIC
c
sof 67.16
10*60*1
13
max
(4-5)
Practically, we chose two parallel connected 10μF capacitors to reduce the ESR value for the
48
prototype circuit experiment.
4-2. Magnetics design
The transformer windings utilized Litz wire to reduce losses. We need to determine the
number of strands for the Litz wire. The formula to calculate the skin depth is:
0
1
f (4-6)
Where σ is the conductivity of the conductor, f is the frequency of the current in the winding,
and μ0 is the permeability of free space [13]. In the circuit to be constructed, μ0 = 4π*10-7 H/m, f
= 80 kHz, and σ =5 .8*107 S/m (for copper wire). Thus, (4-6) can be written as
mmfs
1.66 (4-7)
So, for kHzfs 80 , we calculate the skin depth to be 0.23 mm.
Now we should find breadth b and turns per-section Ns. The concept of the breadth is shown
in Fig 4.1. In Fig 4.1, “b” is the breadth of the winding across the face where one winding faces
another. “Ns” is the number of turns in the section of the winding (In simple windings without
interleaving, Ns=N). Each section shown has a number of turns Ns depending on whether the
different sections of a given winding (primary P or secondary S) are in parallel or series. The
total number of turns N for that winding may be equal to Ns or equal to product of Ns and the
number of sections.
49
Fig 4.1 Examples of the winding breadth b and the number of turns per section [15]
Different strand AWG size and parameters are shown below:
Table 4.1 Parameters for Litz-wire selection[15]
The recommended number of strands for each of the strand diameters being considered is
calculated [15]:
s
eN
bkn
2 (4-8)
In (4-8), k can be found given in Table 4.1 in unit of mm-3. So, b and δ should also be in units of
mm. For coil 1: Ns = 11.21; δ = 0.21mm; b = 10.4mm; k = 130mm3. And ne is determined to be
5.31. The skin depth has already been calculated and the wire we select should have smaller
diameter. As a result, we picked #32 AWG wire and a strand number of 4 to approximately
satisfy the recommended value of ne. The recommended value of ne should be taken as a general
50
guideline. Values of n as much as 25% above or below ne can still be good choices [15]).
Now let’s check whether the designs given by (4-8) will fit in the window space available. To
approximately determine the total area of actual copper, we can calculate the total area of the
copper as NnAs, where N is the number of turns, n is the number of strands, and As is the cross
sectional area of a single strand to estimate the area[14]. The product must be less than 25% to
30% of the window area to be available for the winding. Since we picked n = 4, As = 0.032mm2,
the number of turns for the two inductance (Lr and Lm) are Nlr = (L/Al)^0.5 = 11.21turns and
Nlm = 24.88 turns. So the total area is 1.41 mm2 for Lr and 3.2 mm2 for Lm. The magnetic core of
the inductor was chosen as a Kool Mu toroid one, 0077324A7, produced by Magnetics. Inc. For
the magnetic core 007324A7, AL = 117 +/-8% nH/T2. The window area is 364 mm2 ,the cross
section area Ae = 67.8 mm2. Also, for the magnetic flux density during conduction is:
TAIANB eL
369 10*2.5)10*8.67/(25.0*10*117*12/**
This magnetic flux density value can satisfy the commonly chosen criterion of the saturated
value of 0.25 T. So, the magnetic core has been determined. Thus, the total winding area of the
winding on the right half side of the core is 4.736 mm2, which meets the maximum percentage of
25%.
For the design of the transformer, the magnetic ferrite core in the lab was used. Its type is an
E41-3C90, which has an air gap of approximate 2.59 mm in length and the value of AL to be 220
nH/turn. With the bobbin and Litz wire we have chosen, a transformer is made whose turn ratio
is 4:1:1[16][17][18]. The transformer’s diagram and its equivalent circuit is shown in Fig 4.2. In
Fig 4.2, we should note that the transformer not only has a magnetic inductance that is parallel
51
connected to the primary side but also a leakage inductance that is series connected.
(a) Diagram for a transformer
(b) Equivalent circuit for the transformer in Fig 4.2(a)
Fig 4.2 Diagrams of the transformer[19]
Therefore, the value of Lr is the sum of the inductance of the inductor we construct and the
leakage inductance in the primary side of the transformer. In the lab, we utilized the frequency
response analyzer to measure the values of Lr and Lm. The analyzer is the AP300 model from
Ridley Engineering. The AP300 can measure the value of the inductance and its equivalent
resistance as well.
Firstly, the magnetic inductance Lm is measured. The sweeping frequency range is selected
to be 10 kHz to 100 kHz. The measurement result is shown in Fig 4.3. In the figure, we can
directly see the measurement value of 52.3 μH.
52
Figure 4.3. Magnetic inductance Lm=52.3 µH
Secondly, the leakage inductance of the primary side should be measured. To obtain this
value, the secondary side is short circuited. The resulting measurement is shown in Fig 4.4.
Lm
Equivalent
resistance
53
Figure 4.4. Leakage inductance Ll=3.18 µH
When inductors are connected in series, the equivalent inductance is the sum of each
inductance when the resonant inductor Lr is constructed, we should make its value to be
HLLL leakagerrpractical 25.518.343.8 . the practical value of Lrpractical may have some
deviation from the ideal value. The measurement result is shown in Fig 4.5.
Lleakage
Equivalent
resistance
54
Figure 4.5 Resonant inductance Lr=4.57µH
In the meantime, not only the primary side has leakage inductance, the secondary side also
has leakage inductance. Since the transformer is center-tapped, there should be two leakage
inductance paralleled connected in the circuit. The secondary leakage inductance may not be a
part of consideration during building the transformer, but we should also measure them to
optimize the simulation results. The measurement results are shown in Fig 4.6.
Lrpractical
Equivalent
resistance
55
Figure 4.6 (a). Leakage inductance of the secondary side 1
Figure 4.6 (b). Leakage inductance of the secondary side 2
56
Also, since the resonant capacitor and the filter capacitor have already been selected. The
multi-layer ceramic capacitors produced by Mallory Inc. were simply chosen at the desired value.
Finally, for the load, two fixed value resistors and a variable resistor were picked. The two fixed
value resistors have resistance of 10 Ω and 25 Ω. The variable resistor’s resistance is 3 Ω. The 3
Ω and 10 Ω are series connected, and the 25 Ω one is parallel connected to them.
4-3. Circuit driving and analog control circuit
When the circuit operates properly, the upper MOSFETs in the bridge circuit are floating. In
order to achieve the floating drive of the switches, driving chips are utilized in this circuit to
drive these MOSFETs. With a commonly used analog chip IR 2110, the voltage between the gate
and source can be 10V [20]. As a result, the switches can operate simultaneously with the
feedback signal from the control chip’s output. Thus, the circuit diagram can be redrawn as
shown in Fig 4.7.
Fig 4.7 Circuit diagram of the prototype LLC resonant converter
57
The circuit operating diagram is shown in Fig 4.8
Fig 4.8 Control block diagram
Also, the Miller Effect should be considered. When operating at such high frequency, the
capacitance formed between the gate and drain of MOSFET will affect their operation. To
eliminate the Miller Effect, an additional emitter capacitor is used to shunt the Miller current.
Due to this additional capacitor, the required drive power is increased [21]. The method used to
reduce the Miller Effect is shown in Fig 4.9.
Fig 4.9 Additional capacitor between gate and source [21]
The analog control chip MC34066 (high performance resonant mode controller) is utilized
to regulate the converter’s output voltage. It can be utilized to modulate the switching frequency
through modulated constant on-time or constant off-time control[20]. The upper limit of the
Added
capacitor Cgs’
58
frequency can reach 1MHz. The MC34066’s core components are a reference voltage generator,
a variable frequency oscillator, an error amplifier, a soft-start circuit and a output circuit. The
inner structure of the chip is shown in Fig 4.10.
Fig 4.10 Inner structure of the MC34066 [22]
4-3-1 Reference voltage source
This part of the chip can provide a reference voltage of 5.1 V, and can provide 10 mA of
current to the output as well.
4-3-2 Variable frequency oscillator
The output signals are produced by the oscillator. The circuit contains two
double-threshold comparators, a constant current source and an RC charge-discharge circuit. As
59
seen in Fig. 4.11, when capacitors Cosc and CT are charged, a maximum voltage of 5.1 V is
generated across the two capacitors. The output signal will change states at 3.6 V. Also, the dead
time is determined by the value of RDT, at a range of 0 ns to 800 ns as RDT varies between 0 Ω
and 1 kΩ.
Fig 4.11 Oscillator and one-shot timer [22]
By choosing the oscillator discharge time (tdchg) value and the one shot time, the user can
freely choose the operating mode of the oscillator. Two different modes are shown in Fig 4.8. In
Fig 4.8, when tdchg is bigger than tone-shot, the turn on time in each period is fixed and the turn off
time varies. On the contrary, when tdchg is smaller, the duty cycle is changed due to the variation
of tdchg.
60
Fig 4.12 Timing waveform at 800ns dead time[22]
Capacitors and resistors values used in the circuit are shown in Table 4.2.
Cosc Rosc RVFO RDT CT RT
330pF 68kΩ 100kΩ 1kΩ 22nF 680Ω
Table 4.2 Capacitors and resistors used with the MC 34066
As we’ve already discussed, the method to control the output voltage of the converter is to
change its operating frequency. Thus, PWM is not suitable for control since the frequency of the
PWM waveform is a fixed value. The LLC resonant converter designed operates in pulse
frequency modulation (PFM) mode [23]. Therefore, we should let the chip operates in the mode
of tdchg>tOne-Shot. Pins 7, 8 are input ports of the amplifier. Because the LLC converter operates in
61
Region 2.1, we connect pin 7 to the reference voltage of 5.1 V and connect the feedback voltage
signal to pin 8 to realize negative feedback to the prototype circuit. As a result, the output
frequency signal can be sent to the driving circuit and utilized to adjust the switching frequency
of the converter. The driving and analog control circuits are shown in Fig 4.13.
Fig 4.13 Driving and analog control circuit of the prototype converter
In this chapter, the parameters used for the prototype LLC resonant converter are selected at
first. Then the detailed process of designing magnetic components was illustrated. After the main
circuit of the converter was constructed. The periphery circuits were designed. The driving
circuit formed by the IR 2110 was designed and the Miller effect was reduced. Then the analog
control chip MC 34066 was introduced and its periphery circuit was designed. As a result, the
construction of the hardware circuit was completed. In the next chapter, experimental results will
be shown.
IR 2110
to drive
MOSFE
Ts Da
and Dc
IR 2110
to drive
MOSFE
Ts Db
and Dd
MC 34066
is used to
control the
switching
frequency
Full bridge
formed by
four
MOSFETs
62
CHAPTER 5. Simulation and experimental results
In order to verify the correctness of the theory proposed in the former chapters,
simulation results are presented in this chapter. Meanwhile, a prototype of the LLC resonant
converter was constructed and tested. The parameters of the converter were selected in Chapter 4.
Recall that the converter parameters are:
FCHLHLnFC
VVkHzkHzfVV
fmrr
oin
20;5.52;76.7;470
;7.10;80~60:;40~36:
5-1. Analysis of simulation results
The software SIMPLIS has been used to simulate the operation of the prototype LLC
resonant converter constructed, Fig 5.1 is the simulation schematic. As mentioned earlier, the
practical transformer doesn’t only have leakage inductance on the primary side, but also on the
secondary side. Therefore, leakage inductance values were included in Fig 5.1.
Fig 5.1 Simulation diagram for the LLC resonant converter in SIMPLIS
The simulation result is shown in Fig 5.2. Presented in Fig 5.2 (a) is the resonant current
63
waveform. In the simulation result, we can clearly see the current flowing through Lm increases
linearly until reaching Im. Then the converter moves into the second time interval, and Lm
participates in the resonance. In Fig 5.2 (b), the output voltage is 10.7 V which is the same as the
experimental result. Here we should notice that because the voltage transforming ratio of the
practical transformer is not the idealized ratio, the output voltage of the simulation and
experiment may not be exactly equal to each other.
Fig 5.2 (a). Resonant current during converter operation
64
Fig 5.2 (b). Current in the secondary side of the transformer and the output voltage
Due to the simulation result, the operation of the converter can be verified. Also, we’ve
considered the effect of the leakage inductance of the primary side and made it as a part of the
resonant inductor Lr. Also, the secondary side of the transformer has leakage inductance. When
simulating the converter, two different value of inductors were added to the secondary side to
make the simulation result more practical. That’s the reason why the current flowing through the
Ipeak of D1 is
1.9 A
Ipeak of D2 is
2.3 A
65
diodes D1 and D2 are not exactly the same. The result is shown in Fig 5.2 (b).
5-2. Experimental results
An experimental converter was constructed including the analog control circuitry and the
hardware circuit as shown in Fig 5.3. Because the external resistors and capacitors connected to
the analog control chip MC 34066 are not precisely equal to the calculated values, the output
frequency may have a slight deviation compared to the expected value. The practical operating
frequency range was 60 kHz to 79 kHz, which does not perfectly matching the design range.
Fig 5.3 Prototype of the LLC resonant converter
After powering on the circuit, it operated at the desired frequency at 79 kHz. Using DC
coupling on the oscilloscope, a perturbation was added to the load. As mentioned earlier, the load
was formed by parallel connected resistors. To realize the perturbation, the parallel connected 25
Ω resistor was disconnected. The dynamic response was captured when the disconnection occurs.
In the experiment result, a slight variation in the output voltage is shown in Fig 5.4. The output
voltage returned to 10.7 V.
The
prototype
LLC
resonant
converter
Drive and
control
circuitry
66
Fig 5.4. Output voltage test for the converter
Fig 5.5 shows the resonant current flow in the resonant tank when the converter was
operating. It clearly shows two time intervals as in Fig 5.2 (a).
Fig 5.5 Current flows in resonant tank
Because the ratio of the transformer was 4 and the load and input voltage variation range
was limited, more data is needed to verify the success of the control method. Therefore, several
test results were recorded. Since there are two perturbations (load and input voltage), tests were
Coupling:DC
2.00V/div
Vout:10.7V
Time base: 100μs/div
X:4us/div
Y:1A/div
67
made in two groups. One was changing the load with a fixed input voltage, and another one was
changing the input voltage with a fixed load. The input power can be simply calculated by Vin*Iin.
The output power is calculated by Lo RI *2 . Thus the converter’s efficiency is (Po/Pin)*100%.
Also, another channel is connected to the frequency signal at the output of MC34066. The
ocsilliscope used was a Tektronix MDO3000, which can automatically measure the operating
frequency during operation.
Table 5.1 shows different output voltages and power efficiency at a certain input voltage
value of 36 V.
Output power(W) Switching frequency(kHz) Input current(A) Efficiency
9.16 77.4 0.31 82.08%
10.41 71.8 0.36 80.32%
11.45 68.2 0.40 79.51%
12.48 66.8 0.44 78.78%
14.05 64.9 0.49 79.65%
Table 5.1 Constant input voltage and varying loads
Table 5.2 shows efficiency variations when input voltage was varied and the load was
fixed at 10 Ω. While, the output voltage was fixed at 10.7 V. Thus, the output power was 11.45
W.
68
Input voltage(V) Switching frequency(kHz) Input current(A) efficiency
38.6 78.3 0.34 87.25%
38 69.1 0.36 83.70%
37.5 67 0.38 80.35%
36 64.7 0.40 79.51%
Table 5.2 Constant load and varying input voltages
Steady state tests were captured and recorded by the oscilloscope. Figure 5.6 to Fig 5.9
show the performance of the prototype circuit when either a perturbation occurs in the output
load or input voltage. As mentioned earlier, the perturbation of the output load can be realized by
connecting or disconnecting the parallel connected resistor. The input voltage perturbation is
realized by adjusting the output voltage of the DC voltage source used in lab.
X:200us/div
Y:500mV/div
Load:8.15 Ω to12.5 Ω
69
Fig 5.6 Load change from 8.15 Ω to 12.5 Ω
Fig 5.7 Load changes from 12.5 Ω to 8.15 Ω
Fig 5.8 Input voltage change from 36 V to 38.4 V
X:200us/div
Y:500mV/div
Vin: 36V to 38.4V
X:200us/div
Y:200mV/div
Load: 12.5 Ω to 8.15
Ω
70
Fig 5.9 Input voltage change from 39 V to 37 V
Fig 5.6 to Fig 5.9 shows the result of the steady-state tests of the converter for changes in the
input voltage and load. In these figures, the AC coupling mode was utilized, and the transient
was captured. From these figures, the DC characteristic of the LLC resonant converter can be
verified, along with the PFM control strategy. The output voltage remained fixed at 10.7 V when
perturbations occurred in both the input voltage and load. As a result, the prototype of the LLC
resonant converter has the ability to maintain the steady state output voltage.
X:100us/div
Y:500mV/div
Vin: 39V to 37V
71
CHAPTER 6. Conclusions
Nowadays, LLC resonant converters are utilized in many kinds of occasions, such as a
Data-center, photovoltaic devices and also most commonly seen household appliances like LED
TV sets and microwave-ovens. With more controlling methods proposed and applied, along with
more periphery techniques utilized to optimize the resonant converter, the LLC resonant
converter would be a crucial kind of power electronics technique that contribute to the human
society.
In this thesis, the LLC resonant converter has been discussed in several aspects. Firstly, its
DC characteristics were presented and its operation was described. Then the small signal model
was presented through the method of extended describing functions. In addition, a control
method has been proposed. After theoretical analysis, a prototype converter was constructed.
The analysis and operating region were verified using simulation. The operation of the
converter was verified experimentally. The output voltage remained fixed for changes in the
input voltage and load. In addition, the output ripple was reduced to a satisfied level due to the
selection of the appropriate filter capacitor.
The experimental results may vary a little from the simulation result. However, the converter
circuit obtained the goal of maintaining its output voltage to a certain value when there are
perturbations in both the load and the input voltage.
Also, there are still some future work and suggestions. Firstly, integrated magnetics can be
applied to the circuit [24]. The circuit volume can be significantly reduced with the integrated
magnetic technique utilized. Also, circuit isolation should also be considered. In order to achieve
72
the isolation between the control circuit and the main circuit, a high-speed optocoupler or
isolation transformer can be applied to the circuit. Thirdly, the usage of LLC resonant converters
is not limited to household appliance but also power grid devices as well. Therefore, more
experiments in high-voltage mode should be tested in order to get a more precise conclusion on
its efficiency.
73
BIBLIOGRAPHY
[1] Erickson, Robert W., Dragan Maksimovic, Fundamentals Of Power Electronics. Springer
Science & Business Media, 2001
[2] Bob Mammano, Resonant Mode Converter Topologies, Texas Instrument Technical
Documents, 2001
[3] Bo Yang, “Topology Investigation for Front End DC/DC Power Conversion for
Distributed Power System.” Ph.D. dissertation, Virginia Polytechnic Institute and State
University, 2003.
[4] Vrej Barkhordarian, “Power MOSFET Basics”, International Rectifier Technical
document, El Segundo, CA, Available: www.irf.com/technicalinfo/appnotes/mosfet.pdf.
[5] Arendt Wintrich, Ulrich Nicolar, Werner Turskey, Tobias Reimann. “Power
Semiconductors Application Manual”, SEMIKRON International GmbH, Nov 2010.
[6] Lazar, J.F., Martinelli, R., "Steady-state analysis of the LLC series resonant converter,"
in Applied Power Electronics Conference and Exposition, pp.728-735, vol.2 2001.
[7] Yang, B., Lee, F.C., Zhang, A.J., Guisong Huang, "LLC resonant converter for front end
DC/DC conversion," in Applied Power Electronics Conference and Exposition (APEC),
2002, pp.1108-1112 , vol.2, 2002.
[8] H. Ma, Q. Liu and J. Guo, "A sliding-mode control scheme for llc resonant DC/DC
74
converter with fast transient response," IECON 2012 - 38th Annual Conference on IEEE
Industrial Electronics Society, Montreal, QC, pp. 162-167, 2012.
[9] Yu Fang, Dehong Xu, Zhang Yanjun, Fengchuan Gao, Lihong Zhu, Yi Chen, "Standby
Mode Control Circuit Design of LLC Resonant Converter," in Power Electronics
Specialists Conference, 2007. IEEE, pp.726-730, 17-21 June 2007.
[10] Eric X. Yang. “Extended Describing Function Method for Small-Signal Modeling of
Resonant and Multi-Resonant Converters.” Ph.D. Dissertation, Virginia Polytechnic
Institute and State University, 1994.
[11] Groves, J., "Small-signal analysis using harmonic balance methods," in Power
Electronics Specialists Conference, pp.74-79, 24-27 Jun 1991.
[12] Yang, E.X., Lee, F.C., Jovanovic, M.M., "Small-signal modeling of LCC resonant
converter," in Power Electronics Specialists Conference, pp.941-948 vol.2, 29 Jun-3 Jul
1992.
[13] Clayton R. Paul. Electromagnetic Compatibility, 2nd Edition John Wiley & Sons Inc.
2006.
[14] Sullivan, C.R., "Optimal choice for number of strands in a litz-wire transformer
winding," IEEE Transactions on Power Electronics, vol.14, no.2, pp.283-291, Mar 1999.
[15] Sullivan, C.R., Zhang, R.Y., "Simplified design method for litz wire," in Applied Power
75
Electronics Conference and Exposition (APEC), pp.2667-2674, 16-20 March 2014.
[16] Ferroxcube Inc. Magnetic core E411712 Data Sheet. Sep 2008.
[17] AP Instruments Inc.Model 300 0.01 Hz - 30 MHz Frequency Response Analyzer users’
manual, 2007.
[18] Colonel Wm. T. McLyman, Transformer and Inductor Design Handbook, 4th edition,
Marcel Dekker Inc. 2004.
[19] Mark Halpin, Auburn Univeristy ELEC 5630/6630/6636 Course Note of Electric
Machines, Spring Semester 2015.
[20] Infineon Inc, High and low side driver IR2110 data sheet, 2005.
[21] “Mitigation methods for parasitic turn on effect due to miller capacitor” Avago Technical
Note AV02-0599EN, 2007.
[22] Motolora Inc. Motorola Analog IC Device Data MC34066, 1999.
[23] Zhengmao Zhang, Zhide Tang, "Pulse frequency modulation LLC series resonant X-ray
power supply," in Consumer Electronics, Communications and Networks (CECNet),
pp.1532-1535, 16-18 April 2011.
[24] Yang, B., Chen, R., Lee, F.C., "Integrated magnetic for LLC resonant converter,"
in Applied Power Electronics Conference and Exposition (APEC), pp.346-351 vol.1,
2002
76
APPENDIX. COMPUTER PROGRAM FOR SMALL SIGNAL MODELING
The computer program is utilized to calculate the small signal model of the LLC resonant
converter. The whole computer program is shown in Dr. Eric Yang’s dissertation [10] appendix.
However, the computer program shows the process of calculating SRC resonant converters. The
file “topo.m” should be modified to the LLC resonant converter we’ve designed.
The computer program for defining the LLC resonant converter is shown below:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Name: topo.m -------- LLC topology
% Function: define converter circuit, operating condition,
% and switching boundary condition
% x ------------- current state vector
% u ------------- current input vector
% cur_mode ------ current topological mode
% t ------------- current time
% Output: num_mode ------ # of modes in one cycle
% Para ---------- Circuit Parameters
% x0 ------------ initial condition
% U0 ------------ given input vector
% CTL ----------- Control Parameters
% harm_tbl ------ harmonic table (see Chap.3)
% switching ----- 1 = not cross switching boundary
% -1 = cross switching boundary
% A, B, C, D ---- state matrices of current mode
% Ab,Bb,Cb,Db---- boundary matrices of current mode
%
% Calling: none
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [switching, num_mode, dim_out, harm_tbl, x0, U0, Ts, ...
A, B, C, D, Ab, Bb, Cb, Db] = topo(cur_mode, x, u, t)
%%%%%%%%%%%%%%%%%%% State Equation Description %%%%%%%%%%%%%%%%%%
% Input: U = [Vg, Io];
% Output: Y = [Vo, Ig];
% State: X = [Ilr, Ilm, Vcr, Vcf];
77
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% define # of mode and dimension of output
num_mode = 4;
dim_out = 2;
% define harmonic table
% dc 1st 2nd 3rd
harm_tbl = [0 1 0 1; % Ilr-Ilm -- 1st state
0 1 0 1; % Ilm -- 2nd state
0 1 0 1; % Vcr -- 3nd state
1 0 0 0]; % vo - 4rd state
% define initial condition
x0 = [0; -2; -15; 11]; % [Ilr-Ilm; Ilm; Vcr; vcf]
% define Input variables
U0 = [40; 0]; % [Vg, Io]
% define control variables
Lr = 7.76e-6;
Cr = 470e-9;
Lm = 52.5e-6;
Cf = 20e-6;
rs = 0.01;
rc = 0.01;
Qs = 0.06; % Qs=Zo/R;
Fs = 80e3;
% Some parameters
Zo = sqrt(Lr/Cr); % Zo
Fo = 1/(2*pi*sqrt(Lr*Cr)); % Fo=83337 kHz;
R = Zo / Qs;
% Fsn = Fs / Fo;
k = R / (R+rc);
r = k * rc;
Ts = 1/Fs;
% define switching boundary conditions
if cur_mode == 1,
% set crossing boundary flag
if x(1) < 0.0 ,
switching = -1;
else
switching = 1;
end
% define piecewise linear state equations
A = [(-rs-r)/Lr, -rs/Lr, -1/Lr, -k/Lr;
78
r/Lm, 0, 0, k/Lm;
1/Cr, 1/Cr, 0, 0;
k/Cf, 0, 0, -k/R/Cf];
B = [1/Lr, -r/Lr;
0, r/Lm;
0, 0;
0, k/Cf];
C = [r, 0, 0, k;
1, 1, 0, 0];
D = [0, r;
0, 0];
% switching boundary condition:
% Ab * x + Bb * u + Cb * t + Db < 0
Ab = [1, 0, 0, 0];
Bb = [0, 0];
Cb = 0;
Db = 0;
elseif cur_mode == 2,
if t > 0.5 * Ts;
switching = -1;
else
switching = 1;
end
% define piecewise linear state equations
A = [0, 0, 0, 0;
0, -rs/(Lm+Lr) -1/(Lm+Lr), 0;
0, 1/Cr, 0, 0;
0, 0, 0, -k/R/Cf];
B = [0, 0;
1/(Lm+Lr) 0;
0, 0;
0, k/Cf];
C = [0, 0, 0, k;
0, 1, 0, 0];
D = [0, r;
0, 0];
Ab = [0, 0, 0, 0];
Bb = [0, 0];
Cb = 0;
Db = 0;
elseif cur_mode == 3,
79
if x(1) > 0,
switching = -1;
else
switching = 1;
end,
% define piecewise linear state equations
A = [(-rs-r)/Lr, -rs/Lr, -1/Lr, k/Lr;
r/Lm, 0, 0, -k/Lm;
1/Cr, 1/Cr, 0, 0;
-k/Cf, 0, 0, -k/R/Cf];
B = [-1/Lr, r/Lr;
0, -r/Lm;
0, 0;
0, k/Cf];
C = [-r, 0, 0, k;
-1, -1, 0, 0];
D = [0, r;
0, 0];
Ab = [-1, 0, 0, 0];
Bb = [0, 0];
Cb = 0;
Db = 0;
elseif cur_mode == 4,
if t > Ts,
switching = -1;
else
switching = 1;
end
% define piecewise linear state equations
A = [0, 0, 0, 0;
0, -rs/(Lm+Lr) -1/(Lm+Lr), 0;
00, 1/Cr, 0, 0;
0, 0, 0, -k/R/Cf];
B = [0, 0;
-1/(Lm+Lr), 0;
0, 0;
0, k/Cf];
C = [0, 0, 0, k;
0, -1, 0, 0];
D = [0, r;
0, 0];
80
Ab = [0, 0, 0, 0];
Bb = [0, 0];
Cb = 0;
Db = 0;
end
return;
end
